MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the average velocity of the function over the given interval.
1) y = x2 + 2x, [4, 8]
A) 10

1)

B) 14

C) 7

D) 20

B) - 688

321
C)
4

321
D) 4

2) y = 7x3 + 8x2 - 1, [-8, -4]
A) 688

2)

3) y =

2x, [2, 8]
1

5) y = 4x2 , 0,

3
10

D) 2

7
4

5)

A) 2

1
3

D) -

B) 7

C)

B) -34

1
C) 6

3
10

C)

π
6

D) -

π π
,
4 4

6
π

8)
B)

4
π

C) -

1

4
π

D) 0



1.4 0.49
A) 1

10)

B) 1.5

C) 2

D) 0.5

11) x = 1.
x y
0 0
0.2 0.12
0.4 0.48
0.6 1.08
0.8 1.92
1.0 3
1.2 4.32
1.4 5.88
A) 6

11)

B) 8

C) 4

2

x
y
0.900 -0.05263
0.990 -0.00503
0.999 -0.0005
1.000 0.0000
1.001 0.0005
1.010 0.00498
1.100 0.04762
A) 0.5

B) -0.5

C) 1

D) 0

For the given position function, make a table of average velocities and make a conjecture about the instantaneous
velocity at the indicated time.
14) s(t) = t2 + 8t - 2 at t = 2
14)
t
s(t)

1.9

1.99

1.999


; instantaneous velocity is 5.40
s(t) 5.043 5.364 5.396 5.404 5.436 5.763

3


15) s(t) = t2 - 5 at t = 0

15)

-0.1

t
s(t)

-0.01

-0.001

0.001

0.01

0.1

A)
t -0.1
s(t) -1.4970
-15.0



-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; instantaneous velocity is ∞
-1.5000 -1.4999 -1.4970

t -0.1
s(t) -2.9910
-3.0

-0.01
-2.9999

-0.001
-3.0000

0.001
0.01
0.1
; instantaneous velocity is
-3.0000 -2.9999 -2.9910

B)

17)
D) slope is 13

18) y = x3 - 7x, x = 1
A) slope is -3

B) slope is -4

C) slope is 3

D) slope is 1

19) y = x3 - 2x2 + 4, x = 3
A) slope is 1

B) slope is 0

C) slope is -15

D) slope is 15

20) y = -4 - x3, x = 1
A) slope is 0

B) slope is -1

C) slope is -3

D) slope is 3



21)


22) Given

lim f(x) = Ll, lim f(x) = Lr , and Ll = Lr, which of the following statements is false?
x→0 x→0 +

I.

lim f(x) = Ll
x→0

II.

lim f(x) = Lr
x→0

22)

III. lim f(x) does not exist.
x→0
A) I

B) II

C) III

D) none


D) I and IV only

24) What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x
approaches some value of a?
A) f(a) exists, the limit of f(x) as x→a from the left exists, and the limit of f(x) as x→a from the
right exists.
B) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and
at least one of these limits is the same as f(a).
C) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and
these two limits are the same.
D) Either the limit of f(x) as x→a from the left exists or the limit of f(x) as x→a from the right
exists

5

24)


Use the graph to evaluate the limit.
25) lim f(x)
x→-1

25)

y

1

-6 -5 -4 -3 -2 -1

x→0

26)
y
4
3
2
1

-4

-3

-2

-1

1

2

3

4 x

-1
-2
-3
-4



3

4

5

6 x

-2
-3
-4
-5
-6

A) does not exist

B) 3

C) 0

D) -3

28) lim f(x)
x→0

28)

12


C) does not exist

7

D) 0


29) lim f(x)
x→0

29)

y
4
3
2
1
-4

-3

-2

-1

1

2

3

-2

-1

1

2

3

4 x

-1
-2
-3
-4

A) -1

B) does not exist

C) 1

8

D) ∞


31) lim f(x)
x→0


A) does not exist

B) 0

C) 2

D) -2

32) lim f(x)
x→0

32)
y
4
3
2
1

-4

-3

-2

-1

1

2

2
1

-4

-3

-2

-1

1

2

3

x

4

-1
-2
-3
-4

A) -1
34) Find

B) does not exist


A) -2; -7

B) -5; -2

C) -7; -5

10

D) -7; -2


Use the table of values of f to estimate the limit.
35) Let f(x) = x2 + 8x - 2, find lim f(x).
x→2
x
f(x)

1.9

1.99

1.999

35)

2.001

2.01


x
f(x)

x-4
, find lim f(x).
x-2
x→4
3.9

3.99

3.999

36)

4.001

4.01

4.1

A)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 1.20
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745


37) Let f(x) = x2 - 5, find lim f(x).
x→0
-0.1

x
f(x)

-0.01

37)

-0.001

0.001

0.01

0.1

A)
x -0.1
f(x) -2.9910

-0.01
-2.9999

-0.001
-3.0000


0.001
0.01
0.1
; limit = ∞
-1.5000 -1.4999 -1.4970

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = -15.0
-1.5000 -1.4999 -1.4970

B)

C)

D)

38) Let f(x) =

x

B)
x
4.9
4.99
4.999
5.001
5.01
5.1
; limit = -0.5
f(x) -0.5263 -0.5025 -0.5003 -0.4998 -0.4975 -0.4762
C)
x
4.9
4.99
4.999 5.001
5.01
5.1 ; limit = 0.4
f(x) 0.4263 0.4025 0.4003 0.3998 0.3975 0.3762
D)
x
4.9
4.99
4.999 5.001
5.01
5.1 ; limit = 0.6
f(x) 0.6263 0.6025 0.6003 0.5998 0.5975 0.5762

12



f(x) 0.0304 0.0416 0.0427 0.0430 0.0441 0.0549
B)
x
1.9
1.99
1.999
2.001
2.01
2.1
; limit = -1
f(x) -1.0690 -1.0067 -1.0007 -0.9993 -0.9934 -0.9355
C)
x
1.9
1.99
1.999 2.001
2.01
2.1 ; limit = 0.2429
f(x) 0.2304 0.2416 0.2427 0.2430 0.2441 0.2549
D)
x
1.9
1.99
1.999 2.001
2.01
2.1 ; limit = 0.1429
f(x) 0.1304 0.1416 0.1427 0.1430 0.1441 0.1549
40) Let f(x) =

x

cos (6θ)
, find lim f(θ).
θ
θ→0

x
-0.1
f(θ) -8.2533561

-0.01

-0.001

41)

0.001

0.01

A) limit = 6
C) limit = 8.2533561

0.1
8.2533561

B) limit = 0
D) limit does not exist

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.

f(x)
lim
f(x)
L
x→a
x→a
x→a
x→a

43)

L ≠ 0.
g(x) g(a)
B) lim
, provided that f(a) ≠ 0.
=
f(a)
x→a f(x)
g(x) g(a)
C) lim
.
=
f(a)
x→a f(x)
lim g(x)
x→a
g(x)
M
D) If lim g(x) = M and lim f(x) = L, then lim
=

D) The limit of a constant is the constant, and the limit of a product is the product of the limits.
Find the limit.
46) lim
x→20

10

A) 10

46)
B)

C) 2 5

10

D) 20

47) lim (6x - 4)
x→1
A) -2

47)
C) -10

B) 2

D) 10

48) lim (12 - 10x)

x→2
x→2
x→2
A) -2

51) Let

D) 2

f(x)
lim f(x) = -3 and lim g(x) = 6. Find lim
.
g(x)
x → -5
x → -5
x → -5

A) -

52) Let

C) -80

B) 8

50)

1
2



54) Let lim f(x) = 243. Find lim
x→8
x→8
A) 3

5

C) 53

54)

B) 243

22
5

D) 5

f(x).

55) Let lim f(x) = -9 and lim g(x) = 1. Find lim
x→ 5
x→ 5
x→ 5
A)

53)

C) 8

x→-2
A) -177

58)

56)

57)

B) -113

C) -81

D) -33

x
lim
3x
+2
x→-1
A) -

1
5

58)
B) 1

C) does not exist


4

C) Does not exist

D) -

8
3

61) lim (x + 3)2 (x - 1)3
x→2
A) 1
62) lim
x→2

B) 27

C) 675

D) 25

x2 + 2x + 1

A) 3
63) lim
x→9

61)

62)


63)

64)
B) 2

C) 1/2

D) Does not exist

1+x-1
x

65)
B) Does not exist

C) 1/4

D) 0

Determine the limit by sketching an appropriate graph.
for x < 2
66) lim f(x), where f(x) = -2x - 7
4x
6
for x ≥ 2
x → 2A) -5
67)

68)


for x ≠ -4
for x = -4

68)

B) 13

C) 0

16

D) 19


69)

lim f(x), where f(x) =
x → 4A) 0

70)

lim f(x), where f(x) =
x → -7+
A) -21

4 - x2

0≤x
B) 0

C) 5

D) -1

x4 - 1
72) lim
x→1 x - 1
A) 0

73)

B) Does not exist

lim
x→6

lim
x→6
A)

77)

73)
B) 1

C) 14

D) 7

x→7 x-7
A) Does not exist

74)

72)

75)
B) Does not exist

C) 0

D) 4

x2 + 4x - 60
x2 - 36

4
3

76)
B) -

1
3

C) 0

D) Does not exist


D) Does not exist


79)

lim
h→0

(x + h)3 - x3
h

A) 3x2

80)

79)
B) 3x2 + 3xh + h 2

C) 0

D) Does not exist

6-x
6-x

lim
x→6

80)



D) 1

x2 sin x


; limit = 18.0
f(x) 16.810 17.880 17.988 18.012 18.120 19.210
B)
x
1.9
1.99
1.999 2.001 2.01
2.1
; limit = 17.70
f(x) 16.692 17.592 17.689 17.710 17.808 18.789
C)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = 5.40
f(x) 5.043 5.364 5.396 5.404 5.436 5.763
D)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = ∞
f(x) 5.043 5.364 5.396 5.404 5.436 5.763

18


85) If f(x) =

x
f(x)

x4 - 1

; limit = 5.10
f(x) 4.595 5.046 5.095 5.105 5.154 5.677
D)
x 0.9
0.99 0.999 1.001 1.01 1.1
; limit = 1.210
f(x) 1.032 1.182 1.198 1.201 1.218 1.392

86) If f(x) =

x
f(x)

x3 - 6x + 8
, find lim f(x).
x-2
x→0
-0.1

-0.01

-0.001

86)

0.001

0.01

0.1

0.001
0.01
0.1
-0.01
-0.001
; limit = -2.10
f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574

19


87) If f(x) =

x
f(x)

x-4
, find lim f(x).
x-2
x→4
3.9

3.99

87)

3.999

4.001


f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745
D)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 5.10
f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236
88) If f(x) = x2 - 5, find

x
f(x)

-0.1

lim f(x).
x→0
-0.01

88)

-0.001

0.001

0.01

0.1

-1.5000 -1.4999 -1.4970

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = ∞
-1.5000 -1.4999 -1.4970

x -0.1
f(x) -4.9900

-0.01
-4.9999

-0.001
-5.0000

0.001
0.01
0.1
; limit = -5.0

1.01

1.1

A)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = 2.13640
f(x) 2.15293 2.13799 2.13656 2.13624 2.13481 2.12106
B)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = 0.7071
f(x) 0.72548 0.70888 0.70728 0.70693 0.70535 0.69007
C)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = ∞

4.01

4.1

A)
x
3.9
3.99
3.999
4.001
4.01
4.1
; limit = 0
f(x) -0.02516 -0.00250 -0.00025 0.00025 0.00250 0.02485
B)
x 3.9
f(x) 3.9000

3.99
2.9000

3.999
1.9000

4.001
4.01
4.1
; limit = 1.95
2.0000 3.0000 4.0000


For the function f whose graph is given, determine the limit.
91) Find lim f(x) and
lim f(x).
x→5 x→5 +

91)

y
8
6
4
2
1 2 3 4 5 6 7 8 9 10 11 x

-2 -1
-2
-4
-6
-8

B) ∞, -∞

A) -5, 5
92) Find

lim f(x) and
x→2 5

C) -∞, ∞



5 x

-1
-2
-3
-4
-5

A) ∞; ∞

B) 2; -2

C) 0; 1

22

D) -∞; ∞


93) Find lim f(x).
x→3

93)
5

y

4
3

C) ∞

B) 3

D) does not exist

94) Find lim f(x).
x→-3

94)

6

y

5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1

1

2

3

4

-2

2

4

x

-2

-4

A) ∞

B) 0

C) 1

23

D) -∞


Find the limit.
96)

1
lim
x
x→-2 + 2

D) ∞

lim
tan x
x→(π/2)+

101)
B) ∞

C) 1

D) -∞

lim
sec x
x→(-π/2)-

102)
B) ∞

A) 0
103)

C) -∞

7
lim
2
x → -3- x - 9



1
lim
x
+3
x → -3A) ∞

98)

96)

C) -∞

D) 1

lim (1 + csc x)
x→0+
A) ∞

103)
B) 1

C) 0

D) Does not exist

104) lim (1 - cot x)
x→0
A) ∞



x2 - 5x + 6
x3 - 9x

106)
B) ∞

C) Does not exist

24

D) 0


Find all vertical asymptotes of the given function.
3x
107) f(x) =
x+4
A) x = 4
108) f(x) =

107)

B) x = 3

C) x = -4

x+5
x2 - 64


B) x = 0, x = -25
D) x = 0, x = -5, x = 5

x-1
3
x + 16x

111)

A) x = 0
C) x = 0, x = -16

112) R(x) =

B) x = 0, x = -4, x = 4
D) x = -4, x = 4

-3x2

112)

x2 + 4x - 21

A) x = -7, x = 3
C) x = - 21
113) R(x) =

B) x = -7, x = 3, x = -3
D) x = 7, x = -3



7
,x=1
2

C) x =

2
, x = -1
7

x-3
9x - x3

D) x =

7
, x = -1
2

115)

A) x = -3, x = 3
C) x = 0, x = -3

B) x = 0, x = -3, x = 3
D) x = 0, x = 3

25